i finally have the mathematica program working (at least it seems to be... might need a bit more testing still) that demonstrates how the geometry of a ball rolling on another of 3 times the radius is equivalent to the null subalgebra geometry of the split octonions.

actually, after that bit more of testing, i think that it's still not working perfectly yet. but the basic plan of starting from the most special bruhat classes and working towards the most generic does seem to be working; the correspondence seems to be straightened out for the special classes but not yet for the most generic one.